{"id":74,"date":"2021-07-30T17:09:08","date_gmt":"2021-07-30T17:09:08","guid":{"rendered":"https:\/\/courses.lumenlearning.com\/calculus3\/?post_type=chapter&#038;p=74"},"modified":"2022-10-29T01:05:53","modified_gmt":"2022-10-29T01:05:53","slug":"summary-of-motion-in-space","status":"publish","type":"chapter","link":"https:\/\/courses.lumenlearning.com\/calculus3\/chapter\/summary-of-motion-in-space\/","title":{"raw":"Summary of Motion in Space","rendered":"Summary of Motion in Space"},"content":{"raw":"<div class=\"textbox learning-objectives\">\r\n<h3>Essential Concepts<\/h3>\r\n<div class=\"PageContent-ny9bj0-0 gLsPXs\" tabindex=\"0\">\r\n<div id=\"main-content\" class=\"MainContent__HideOutline-sc-6yy1if-0 bdVAq sc-bdVaJa ibTDPh\" tabindex=\"-1\" data-dynamic-style=\"false\">\r\n<div id=\"composite-page-11\" class=\"os-eoc os-key-concepts-container\" data-type=\"composite-page\" data-uuid-key=\".key-concepts\">\r\n<div class=\"os-key-concepts\">\r\n<div class=\"os-section-area\"><section id=\"fs-id1169737388310\" class=\"key-concepts\" data-depth=\"1\">\r\n<ul id=\"fs-id1169737388317\" data-bullet-style=\"bullet\">\r\n \t<li>If\u00a0[latex]{\\bf{r}}(t)[\/latex]\u00a0represents the position of an object at time [latex]t[\/latex], then\u00a0[latex]{\\bf{r}}^{\\prime}(t)[\/latex]\u00a0represents the velocity and\u00a0[latex]{\\bf{r}}^{\\prime\\prime}(t)[\/latex]\u00a0represents the acceleration of the object at time\u00a0[latex]t[\/latex].\u00a0The magnitude of the velocity vector is speed.<\/li>\r\n \t<li>The acceleration vector always points toward the concave side of the curve defined by [latex]{\\bf{r}}(t)[\/latex]. The tangential and normal components of acceleration [latex]a_{\\bf{T}}[\/latex] and [latex]a_{\\bf{N}}[\/latex] are the projections of the acceleration vector onto the unit tangent and unit normal vectors to the curve.<\/li>\r\n \t<li>Kepler\u2019s three laws of planetary motion describe the motion of objects in orbit around the Sun. His third law can be modified to describe motion of objects in orbit around other celestial objects as well.<\/li>\r\n \t<li>Newton was able to use his law of universal gravitation in conjunction with his second law of motion and calculus to prove Kepler\u2019s three laws.<\/li>\r\n<\/ul>\r\n<\/section><\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<\/div>\r\n<div class=\"PrevNextBar__BarWrapper-sc-13m2i12-3 fEZPiF\" data-analytics-region=\"prev-next\"><\/div>\r\n<\/div>\r\n<h2>Key Equations<\/h2>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Velocity\r\n<\/strong>[latex]{\\bf{v}}(t)={\\bf{r}}^{\\prime}(t)[\/latex]<\/li>\r\n \t<li><strong>Acceleration<\/strong>\r\n[latex]{\\bf{a}}(t)={\\bf{v}}^{\\prime}(t)={\\bf{r}}^{\\prime\\prime}(t)[\/latex]<\/li>\r\n \t<li><strong>Speed<\/strong>\r\n[latex]v(t)=\\parallel{\\bf{v}}(t)\\parallel=\\parallel{\\bf{r}}^{\\prime}(t)\\parallel=\\frac{ds}{dt}[\/latex]<\/li>\r\n \t<li><strong>Tangential component of acceleration<\/strong>\r\n[latex]a_{\\bf{T}}={\\bf{a}}\\cdot{\\bf{T}}=\\frac{\\bf{v}\\cdot\\bf{a}}{\\parallel{\\bf{v}}\\parallel}[\/latex]<\/li>\r\n<\/ul>\r\n<ul id=\"fs-id1169736654976\">\r\n \t<li><strong>Normal component of acceleration<\/strong>\r\n[latex]a_{\\bf{N}}={\\bf{a}}\\cdot{\\bf{N}}=\\frac{\\parallel{\\bf{v}}\\times{\\bf{a}}\\parallel}{\\parallel{\\bf{v}}\\parallel}=\\sqrt{\\parallel{\\bf{a}}\\parallel^{2}-{a_{\\bf{T}}}^{2}}[\/latex]<\/li>\r\n<\/ul>\r\n<h2>Glossary<\/h2>\r\n<dl class=\"definition\">\r\n \t<dt>acceleration vector<\/dt>\r\n \t<dd>the second derivative of the position vector<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>Kepler's laws of planetary motion<\/dt>\r\n \t<dd>three laws governing the motion of planets, asteroids, and comets in orbit around the Sun<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>normal component of acceleration<\/dt>\r\n \t<dd>the coefficient of the unit normal vector [latex]{\\bf{N}}[\/latex]<span class=\"Apple-converted-space\">\u00a0<\/span>when the acceleration vector is written as a linear combination of<span class=\"Apple-converted-space\">\u00a0[latex]{\\bf{T}}[\/latex]\u00a0and\u00a0[latex]{\\bf{N}}[\/latex]<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>projectile motion<\/dt>\r\n \t<dd>motion of an object with an initial velocity but no force acting on it other than gravity<\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>tangential component of acceleration<\/dt>\r\n \t<dd>the coefficient of the unit tangent vector [latex]{\\bf{T}}[\/latex]<span style=\"font-size: 1em;\"> when the acceleration vector is written as a linear combination of\u00a0[latex]{\\bf{T}}[\/latex] and [latex]{\\bf{N}}[\/latex]<\/span><\/dd>\r\n<\/dl>\r\n<dl class=\"definition\">\r\n \t<dt>velocity vector<\/dt>\r\n \t<dd>the derivative of the position vector<\/dd>\r\n<\/dl>","rendered":"<div class=\"textbox learning-objectives\">\n<h3>Essential Concepts<\/h3>\n<div class=\"PageContent-ny9bj0-0 gLsPXs\" tabindex=\"0\">\n<div id=\"main-content\" class=\"MainContent__HideOutline-sc-6yy1if-0 bdVAq sc-bdVaJa ibTDPh\" tabindex=\"-1\" data-dynamic-style=\"false\">\n<div id=\"composite-page-11\" class=\"os-eoc os-key-concepts-container\" data-type=\"composite-page\" data-uuid-key=\".key-concepts\">\n<div class=\"os-key-concepts\">\n<div class=\"os-section-area\">\n<section id=\"fs-id1169737388310\" class=\"key-concepts\" data-depth=\"1\">\n<ul id=\"fs-id1169737388317\" data-bullet-style=\"bullet\">\n<li>If\u00a0[latex]{\\bf{r}}(t)[\/latex]\u00a0represents the position of an object at time [latex]t[\/latex], then\u00a0[latex]{\\bf{r}}^{\\prime}(t)[\/latex]\u00a0represents the velocity and\u00a0[latex]{\\bf{r}}^{\\prime\\prime}(t)[\/latex]\u00a0represents the acceleration of the object at time\u00a0[latex]t[\/latex].\u00a0The magnitude of the velocity vector is speed.<\/li>\n<li>The acceleration vector always points toward the concave side of the curve defined by [latex]{\\bf{r}}(t)[\/latex]. The tangential and normal components of acceleration [latex]a_{\\bf{T}}[\/latex] and [latex]a_{\\bf{N}}[\/latex] are the projections of the acceleration vector onto the unit tangent and unit normal vectors to the curve.<\/li>\n<li>Kepler\u2019s three laws of planetary motion describe the motion of objects in orbit around the Sun. His third law can be modified to describe motion of objects in orbit around other celestial objects as well.<\/li>\n<li>Newton was able to use his law of universal gravitation in conjunction with his second law of motion and calculus to prove Kepler\u2019s three laws.<\/li>\n<\/ul>\n<\/section>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<\/div>\n<div class=\"PrevNextBar__BarWrapper-sc-13m2i12-3 fEZPiF\" data-analytics-region=\"prev-next\"><\/div>\n<\/div>\n<h2>Key Equations<\/h2>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Velocity<br \/>\n<\/strong>[latex]{\\bf{v}}(t)={\\bf{r}}^{\\prime}(t)[\/latex]<\/li>\n<li><strong>Acceleration<\/strong><br \/>\n[latex]{\\bf{a}}(t)={\\bf{v}}^{\\prime}(t)={\\bf{r}}^{\\prime\\prime}(t)[\/latex]<\/li>\n<li><strong>Speed<\/strong><br \/>\n[latex]v(t)=\\parallel{\\bf{v}}(t)\\parallel=\\parallel{\\bf{r}}^{\\prime}(t)\\parallel=\\frac{ds}{dt}[\/latex]<\/li>\n<li><strong>Tangential component of acceleration<\/strong><br \/>\n[latex]a_{\\bf{T}}={\\bf{a}}\\cdot{\\bf{T}}=\\frac{\\bf{v}\\cdot\\bf{a}}{\\parallel{\\bf{v}}\\parallel}[\/latex]<\/li>\n<\/ul>\n<ul id=\"fs-id1169736654976\">\n<li><strong>Normal component of acceleration<\/strong><br \/>\n[latex]a_{\\bf{N}}={\\bf{a}}\\cdot{\\bf{N}}=\\frac{\\parallel{\\bf{v}}\\times{\\bf{a}}\\parallel}{\\parallel{\\bf{v}}\\parallel}=\\sqrt{\\parallel{\\bf{a}}\\parallel^{2}-{a_{\\bf{T}}}^{2}}[\/latex]<\/li>\n<\/ul>\n<h2>Glossary<\/h2>\n<dl class=\"definition\">\n<dt>acceleration vector<\/dt>\n<dd>the second derivative of the position vector<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>Kepler&#8217;s laws of planetary motion<\/dt>\n<dd>three laws governing the motion of planets, asteroids, and comets in orbit around the Sun<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>normal component of acceleration<\/dt>\n<dd>the coefficient of the unit normal vector [latex]{\\bf{N}}[\/latex]<span class=\"Apple-converted-space\">\u00a0<\/span>when the acceleration vector is written as a linear combination of<span class=\"Apple-converted-space\">\u00a0[latex]{\\bf{T}}[\/latex]\u00a0and\u00a0[latex]{\\bf{N}}[\/latex]<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>projectile motion<\/dt>\n<dd>motion of an object with an initial velocity but no force acting on it other than gravity<\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>tangential component of acceleration<\/dt>\n<dd>the coefficient of the unit tangent vector [latex]{\\bf{T}}[\/latex]<span style=\"font-size: 1em;\"> when the acceleration vector is written as a linear combination of\u00a0[latex]{\\bf{T}}[\/latex] and [latex]{\\bf{N}}[\/latex]<\/span><\/dd>\n<\/dl>\n<dl class=\"definition\">\n<dt>velocity vector<\/dt>\n<dd>the derivative of the position vector<\/dd>\n<\/dl>\n\n\t\t\t <section class=\"citations-section\" role=\"contentinfo\">\n\t\t\t <h3>Candela Citations<\/h3>\n\t\t\t\t\t <div>\n\t\t\t\t\t\t <div id=\"citation-list-74\">\n\t\t\t\t\t\t\t <div class=\"licensing\"><div class=\"license-attribution-dropdown-subheading\">CC licensed content, Shared previously<\/div><ul class=\"citation-list\"><li>Calculus Volume 3. <strong>Authored by<\/strong>: Gilbert Strang, Edwin (Jed) Herman. <strong>Provided by<\/strong>: OpenStax. <strong>Located at<\/strong>: <a target=\"_blank\" href=\"https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction\">https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/a>. <strong>License<\/strong>: <em><a target=\"_blank\" rel=\"license\" href=\"https:\/\/creativecommons.org\/licenses\/by-nc-sa\/4.0\/\">CC BY-NC-SA: Attribution-NonCommercial-ShareAlike<\/a><\/em>. <strong>License Terms<\/strong>: Access for free at https:\/\/openstax.org\/books\/calculus-volume-3\/pages\/1-introduction<\/li><\/ul><\/div>\n\t\t\t\t\t\t <\/div>\n\t\t\t\t\t <\/div>\n\t\t\t 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